No you’re not. Math’s infrastructure is not designed for PDE.
I’m glad you’re dropping STALD for the moment. The original STALD paper(I think it’s in the reference of cvode user guide) shows that it’s designed for specific problems (advection-convection) where spurious oscillations could happen when certain BDF methods are used, so it will unlikely benefit problems like the heat equation @jtimonen posted earlier, and most flow solvers nowadays no longer practice this anyway. In addition, by assuming small problems size and using serial vector, Stan’s CVODES and adjoint implementation becomes irrelevant for most real-world PDE problems. Again, it comes down to infrastructure.
IMO the best bet is to interfacing external solvers, as we’ve seen here.